Question

Let
$\alpha=\displaystyle\sum_{ k =1}^{\infty} \sin ^{2 k }\left(\frac{\pi}{6}\right)$
Let $g:[0,1] \rightarrow R$ be the function defined by
$g(x)=2^{\alpha x}+2^{\alpha(1-x)}$
Then, which of the following statements is/are TRUE?

• A. The minimum value of $g(x)$ is $2^{\frac{7}{6}}$
• B. The maximum value of $g(x)$ is $1+2^{\frac{1}{3}}$
• C. The function $g( x )$ attains its maximum at more than one point
• D. The function $g( x )$ attains its minimum at more than one point

Answer 1

Answer: (A), (B), (C)

$ \alpha=\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^4+\left(\frac{1}{2}\right)^6+\ldots$
$ \alpha=\frac{\frac{1}{4}}{1-\frac{1}{4}}=\frac{1}{3} $
$ \therefore g ( x )=2^{ x / 3}+2^{1 / 3(1- x )} $
$ \therefore g ( x )=2^{ x / 3}+\frac{2^{1 / 3}}{2^{x / 3}} $
where $g(0)=1+2^{1 / 3} \& g (1)=1+2^{1 / 3}$
$\therefore g ^{\prime}( x )=\frac{1}{3}\left(2^{ x / 3}-\frac{2^{1 / 3}}{2^{x / 3}}\right)=0 $
$ \Rightarrow 2^{2 x / 3}=2^{1 / 3} \Rightarrow x =\frac{1}{2}=$ critical point